Central limit theorem

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Beyond the classical framework

Log-concave Joint density Total variation distance Uncorrelated Pairwise independent

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History

Henk Tijms Abraham de Moivre Pierre-Simon Laplace Aleksandr Lyapunov Francis Galton

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Remarks

Law of large numbers Mean Taylor's theorem Little o notation Exponential function

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Independent sequences

Sample average Law of large numbers Converges almost surely Converges in probability Converge in distribution

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Field: Probability theory
Generalizations: Lindeberg's CLT
Statement: The scaled sum of a sequence of i.i.d. random variables with finite positive variance converges in distribution to the normal distribution.
Type: Theorem

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Overview

Probability theory
Probability distribution
Normally distributed Normal distribution
Statistics
Statistical sample Sampling (statistics)
Expected value
Variance
Random variable
Limit as n → ∞ {\displaystyle n\to \infty } of the distribution Convergence of random variables
Observations Random variate
Arithmetic mean
Independent and identically distributed
Binomial distribution
De Moivre–Laplace theorem
Density Probability density function
Convolution
Independent and identically distributed random variables
Characteristic function Characteristic function (probability theory)
Fourier transforms Fourier transform

Independent sequences

Sample average Sample mean
Law of large numbers
Converges almost surely Almost sure convergence
Converges in probability Convergence in probability
Converge in distribution Convergence in distribution
Cumulative distribution functions Cumulative distribution function
Supremum
Moments Moment (mathematics)
Lindeberg Jarl Waldemar Lindeberg
Indicator function
Random vector
Covariance matrix
Multivariate normal distribution
Cramér–Wold theorem
Berry–Esseen Berry–Esseen theorem

The generalized central limit theorem

Dependent processes

Strong mixing coefficient Mixing (mathematics)
Stationary sequence
Bradley 2007 Central limit theorem
Martingale Martingale (probability theory)

Remarks

Law of large numbers Proof of the law of large numbers
Mean Linearity of expectation
Taylor's theorem
Little o notation Little-o notation
Exponential function
Lévy's continuity theorem Lévy continuity theorem
Asymptotic distribution
Uniform Uniform convergence
Stein's method
Entropy Information entropy
Monotonically Monotonic function
Discrete random variables Discrete random variable
Histogram
Iid
Sampling distribution
Glivenko–Cantelli theorem
Z-test
Asymptotic series
Cauchy distribution
Slowly varying function
Law of the iterated logarithm

Extensions

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Map overview Semantic statistics

Nodes120

Edges119

Triples175

Avg. degree1.98

Density0.016667

Components1

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Central limit theorem

Top relations

related to References · 76

Central limit theorem → Annals, Archived, ASA, Basic Properties, Bauer, Berlin, Berry, Bibcode, Billingsley, Bo'az, Bradley, Bárány, Cambridge University Press, Central, Christou, Demonstration Activity, Dinov, Durrett, Esseen, Fischer

see also · 13

Central limit theorem → Asymptotic, Berry, Central, CLT, Erdős, Esseen, Gnedenko, Hall, Irwin, Kac, Poisson, Tippett, Xn

related to The generalized central limit theorem · 11

Central limit theorem → An English, Andrey Kolmogorov, Boris Vladimirovich Gnedenko, French, GCLT, Jarl Waldemar Lindeberg, Kolmogorov's, Paul Lévy, Sergei Bernstein, The, William Feller

related to External links · 10

Central limit theorem → Carl McTague, Central, EMS Press, Encyclopedia, Eric, Galton, Khan Academy, Mathematics, MathWorld, Weisstein

related to Calculating the variance · 8

Central limit theorem → Construct, In, Lemma, Let Sn, Many, Sn, Suppose, Var

related to Common misconceptions · 8

Central limit theorem → Cantelli, Glivenko, In, See Z-test, Studies, That, The, These

related to Convergence to the limit · 6

Central limit theorem → As, Berry, Esseen, If, Stein's, The

related to Lyapunov CLT · 6

Central limit theorem → Define, In, Lyapunov, Lyapunov CLT, Suppose, The

related to Products of positive random variables · 6

Central limit theorem → Gibrat's, Many, The, Therefore, This, Whereas

related to Multidimensional CLT · 5

Central limit theorem → For, Proofs, Sigma, Summation, The

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Important terminology

theorem limit distribution random displaystyle central normal variables textstyle mean sum frac mu sigma variance probability sqrt right independent left

Entity relationships Subject–Predicate–Object triples

Subject	Predicate	Object	Confidence	Src
Central limit theorem	Field	Probability theory	1.00	infobox
Central limit theorem	Generalizations	Lindeberg's CLT	1.00	infobox
Central limit theorem	Statement	The scaled sum of a sequence of i.i.d. random variables with finite positive variance converges in distribution to the normal distribution.	1.00	infobox
Central limit theorem	Type	Theorem	1.00	infobox
Central limit theorem	has application	The	0.60	section
Central limit theorem	has application	Since	0.60	section
Central limit theorem	has application	It	0.60	section
Central limit theorem	related to Calculating the variance	Let Sn	0.60	section
Central limit theorem	related to Calculating the variance	Many	0.60	section
Central limit theorem	related to Calculating the variance	Sn	0.60	section
Central limit theorem	related to Calculating the variance	Var	0.60	section
Central limit theorem	related to Calculating the variance	In	0.60	section

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Beyond the classical framework

History

Remarks

Independent sequences

Key facts & relationships

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Overview

Independent sequences

The generalized central limit theorem

Dependent processes

Remarks

Extensions

Beyond the classical framework

Applications and examples

History

Advanced semantic analysis

Map overview Semantic statistics

Central limit theorem

How this topic connects Entity context

Central limit theorem

Top relations

Important terminology Word statistics

Important terminology

Entity relationships Subject–Predicate–Object triples

Related concept clusters Concept neighborhoods

Connections between topic areas Semantic bridges